Overstable Convective Modes of Rotating Hot Jupiters

We calculate overstable convective modes of uniformly rotating hot Jupiters, which have a convective core and a thin radiative envelope. Convective modes in rotating planets have complex frequency $\omega$ and are stabilized by rapid rotation so that their growth rates $\propto\omega_{\rm I}={\rm Im}(\omega)$ are much smaller than those for the non-rotating planets. The stabilized convective modes excite low frequency gravity waves in the radiative envelope by frequency resonance between them. We find that the convective modes that excite envelope gravity waves remain unstable even in the presence of non-adiabatic dissipations in the envelope. We calculate the heating rates due to non-adiabatic dissipations of the oscillation energy of the unstable convective modes and find that the magnitudes of the heating rates cannot be large enough to inflate hot Jupiters sufficiently so long as the oscillation amplitudes remain in the linear regime.